Does dark energy or cosmic inflation explain flatness?

[RogerWaters]

TL;DR Summary

Inflation is sometimes presented as explaining why the universe has flat geometry; but so is the 'missing 70%' mass-energy density of dark energy.

As I understand, the main theoretical virtue of Guth's inflation hypothesis is that it explains a bunch of otherwise hard-to-account-for phenomena under the standard big bang model without inflation: the Horizon Problem, the Flatness problem, the Monopole problem, and also the problem of how small fluctuations in the CMB got there that would later result in the non-uniform gravitational collapse of matter into the structures we see today in the universe (against the background of a largely homogenous universe, of course).

So cosmic inflation explains flatness, among other things. However, I also read that the total mass-energy density of the universe explains flatness, too. Before the discovery that the expansion rate of the universe is increasing in the late 1990s, weighing the universe by measuring the mass of galaxies and clusters and even taking into account dark matter produced a figure that was 30 percent of what was required to produce a flat universe. There was a 'missing' 70 percent.

Then it was discovered that the rate at which the universe is expanding is increasing. The amount of energy that space would have to contain in order to produce the observed accelerating expansion was, remarkably, the missing 70 percent mass-energy needed to make a flat universe.

So dark energy explains flatness.

But now it seems that flatness is 'over explained'. It is explained by both cosmic inflation and dark energy. A natural conclusion a layperson like me would arrive at from this is that cosmic inflation and dark energy are the same thing: natural for the reason that both inflation and dark energy seem to involve a rather mysterious expansion of empty space itself. However, i read that, while inflation and dark energy are sometimes described as 'cousins', they are not currently understood as one and the same phenomena.

So what actually explains flatness?

 

[kimbyd]

Dark energy doesn't explain flatness. It's only a late-time effect that would have reduced observed spatial curvature by a small amount.

The flatness problem is a problem only in the early universe. For the first few billion years, the effect of spatial curvature increased over time. This means that in order for the effective curvature today to close to flat, it had to be vastly closer to flat in the early universe. So instead of having a number like 0.01 (today's curvature fraction is somewhere less than this), in the distant past you would have needed something like $10^{-6}$. Small numbers like that demand an explanation, and cosmic inflation may be one way to solve this problem.

Modifying the expansion a little at late times (dark energy) doesn't really have a noticeable impact.

 

[PeterDonis]

The amount of energy that space would have to contain in order to produce the observed accelerating expansion was, remarkably, the missing 70 percent mass-energy needed to make a flat universe.

What you state here is not a solution to the flatness problem; it is the flatness problem.

The flatness problem is that, unless the universe has exactly the critical density always, during any phase that is radiation or matter dominated (as the universe was from shortly after the Big Bang until a few billion years ago), it will evolve away from flatness. So without either some extraordinary fine tuning of initial conditions (to put the universe at critical density to something like 24 decimal places, IIRC), or something like inflation to create flatness prior to the reheating at the Big Bang, we would not expect today's universe to be flat or even close to flat. (As @kimbyd says, the dark energy dominated phase during the last few billion years does not change that significantly; the current density of dark energy would take far, far longer than the age of the universe to get it close to flatness by itself.)

 

[RogerWaters]

What you state here is not a solution to the flatness problem; it is the flatness problem.

The flatness problem is that, unless the universe has exactly the critical density always, during any phase that is radiation or matter dominated (as the universe was from shortly after the Big Bang until a few billion years ago), it will evolve away from flatness. So without either some extraordinary fine tuning of initial conditions (to put the universe at critical density to something like 24 decimal places, IIRC), or something like inflation to create flatness prior to the reheating at the Big Bang, we would not expect today's universe to be flat or even close to flat. (As @kimbyd says, the dark energy dominated phase during the last few billion years does not change that significantly; the current density of dark energy would take far, far longer than the age of the universe to get it close to flatness by itself.)

Thanks Peter,

In that case, before the discovery of accelerating expansion, why is the 70 percent mass-energy contributed by this dark energy described as 'missing': missing in relation to the universe being flat? If cosmic inflation happened, wouldn't it have pushed any initially-curved amount of mass-energy towards flatness? Why is a certain amount of mass-energy often described* as being 'necessary' for flatness if it's cosmic inflation that does the work?

*Here is Krauss on page 86 of A Universe from Nothing: "[70 percent] is, remarkably, what is needed in order to make a flat universe consistent with the fact that only 30 percent of the required mass exists in and around galaxies and clusters". Here is Brian Greene in The Fabric of the Cosmos: "Since the early days of general relativity, physicists have realized that the total matter and energy in each volume of space - the matter/energy density - determine the curvature of space... for a very special amount of matter/energy density - the critical density - space will... be perfectly flat: that is, there will be no curvature". Here is Ostriker and Mitton in Heart of Darkness: Unravelling the Mysteries of the Invisible Universe: "Dark energy fills the gap, allowing the flat, 'Just right' universe... the sum of matter (mostly dark matter) and the dark energy did total the amount required to produce a geometrically flat universe". (I understand these are not peer-reviewed journal articles but I can't understand those when it comes to cosmology).

 

[kimbyd]

Thanks Peter,

In that case, before the discovery of accelerating expansion, why is the 70 percent mass-energy contributed by this dark energy described as 'missing': missing in relation to the universe being flat? If cosmic inflation happened, wouldn't it have pushed any initially-curved amount of mass-energy towards flatness? Why is a certain amount of mass-energy often described* as being 'necessary' for flatness if it's cosmic inflation that does the work?

*Here is Krauss on page 86 of A Universe from Nothing: "[70 percent] is, remarkably, what is needed in order to make a flat universe consistent with the fact that only 30 percent of the required mass exists in and around galaxies and clusters". Here is Brian Greene in The Fabric of the Cosmos: "Since the early days of general relativity, physicists have realized that the total matter and energy in each volume of space - the matter/energy density - determine the curvature of space... for a very special amount of matter/energy density - the critical density - space will... be perfectly flat: that is, there will be no curvature". (I understand these are not peer-reviewed journal articles but I can't understand those when it comes to cosmology).

Flatness implies that the mass/energy density balances the rate of expansion. When we look at the amount of matter (dark+normal), we only get something like 30% of what is needed to balance the observed rate of expansion.

This is perhaps most obvious when it comes to the CMB data combined with local measures of the rate of expansion. The CMB was emitted when the universe transitioned from an opaque plasma to a transparent gas. Because the entire universe was illuminated, we get a very good picture of how much matter there was at that time. Getting an understanding of how much normal matter there was at that time is perhaps obvious, but we also get a very accurate estimate of how much dark matter there was. I won't go into detail of how just now, though.

With a very clear picture of the total matter density at the time the CMB was emitted, we can understand how much the universe has expanded by looking at the temperature of the CMB: the temperature at the time of emission is set by when a hydrogen-helium plasma of that density transitions from a plasma to a gas (roughly 3000K). The current, observed temperature gives us a precise estimate of how much expansion there has been (about a factor of 1090).

So we know what the density was at that time, and we know how much the universe has expanded. That tells us the average density today. When we compare that measurement with the rate of expansion today, we get two things:
1) The image of the CMB itself looks like we're viewing it through a spatially-flat universe.
2) The matter density is less than 1/3rd of what it needs to be to balance the observed rate of expansion.

The simplest model for what could explain this is a cosmological constant. The value of the cosmological constant is very weird, but it definitely fits the data.

 

[PeterDonis]

before the discovery of accelerating expansion, why is the 70 percent mass-energy contributed by this dark energy described as 'missing': missing in relation to the universe being flat?

The universe was observed to be spatially flat decades before dark energy was discovered. Even leaving aside all the issues I mentioned, if the universe is in fact spatially flat, or so close to it that we can't tell the difference, then the total energy density must be the critical density, or so close to it that we can't tell the difference. And "30% of critical density" certainly does not meet that requirement. So before dark energy was discovered, there was a huge disconnect between our observations of spatial flatness and our observations of energy density. The discovery of dark energy solved that disconnect, but left other questions still unresolved.

 

[Bandersnatch]

So before dark energy was discovered, there was a huge disconnect between our observations of spatial flatness and our observations of energy density.

Was this, historically, an actual problem on people's minds? Riess' work was before WMAP and PLANCK, after all. I think there was only COBE probing the early universe before then, and I'm not sure if there was good enough data available to start worrying about this.
I wonder if this might be a somewhat anachronistic projection of modern advances onto that period.

 

[PeterDonis]

Was this, historically, an actual problem on people's minds?

I believe so, yes. Otherwise you wouldn't have physicists like Krauss saying things like those the OP quoted.

I think there was only COBE probing the early universe before then, and I'm not sure if there was good enough data available to start worrying about this.

AFAIK the fact that the universe is close to spatially flat has been known for decades, as I said before. Certainly we now know it to much greater accuracy, but I don't think there's ever been a time where the error bars on the observed spatial curvature were anywhere close to being consistent with the (strongly negative) value that would be expected from a density only 30% of critical (let alone a density of 5% of critical or so, which is all that ordinary matter, i.e., all the matter we can actually see, can account for).

 

[RogerWaters]

Thanks for clearing that up guys.

The mass/energy density of the universe needs to be a certain number because we know the universe is flat. Without dark energy that density was measured to only around 30 percent of the needed value. Dark energy was then discovered and, remarkably, made up the other 70 percent.

TL;DR: The critical mass/energy density of the universe, to which dark energy contributes 70 percent, is a by-product of flatness not a cause.

I then wonder why these kinds of statements are made (in this case by Greene) in the context of the overall shape of space: "Since the early days of general relativity, physicists have realized that the total matter and energy in each volume of space - the matter/energy density - determine the curvature of space."

I get that mass/energy can warp spacetime on a local level, but on a large scale, if inflation solved the flatness problem it was inflation that determined the curvature of space not matter/energy density. The matter/energy density is an effect not a cause of the overall shape of the universe.

 

[PeterDonis]

The critical mass/energy density of the universe, to which dark energy contributes 70 percent, is a by-product of flatness not a cause.

No, this is not correct. The critical density causes spatial flatness, not the other way around. The fact that we observed the flatness first, before we observed all of the components of the density that add up to the critical density, does not mean the causation has to run that way.

I then wonder why these kinds of statements are made (in this case by Greene) in the context of the overall shape of space: "Since the early days of general relativity, physicists have realized that the total matter and energy in each volume of space - the matter/energy density - determine the curvature of space."

Because it's true; that's how the Einstein Field Equation works. Stress-energy, which is what "the total matter and energy in each volume of space" refers to, determines the geometry of spacetime, which includes "the curvature of space", via that equation.

Note that "the curvature of space" (or lack thereof, in this case) has to be carefully interpreted, because "space" is frame-dependent. The "space" being referred to is the "space" seen by all comoving observers at an instant of cosmological time. These observers are picked out by the symmetry of the spacetime, which is why this is the "space" that cosmologists standardly refer to.

I get that mass/energy can warp spacetime on a local level, but on a large scale, if inflation solved the flatness problem it was inflation that determined the curvature of space not matter/energy density.

Wrong. "Inflation" is just a shorthand name for a model that is dominated by a particular kind of stress-energy, which is a scalar field with a huge vacuum energy density. It's that property that causes inflationary expansion.

The matter/energy density is an effect not a cause of the overall shape of the universe.

No. See above.

 

[George Jones]

The universe was observed to be spatially flat decades before dark energy was discovered.

By whom? Dark energy was discovered in 1998 using Type 1a supernovae as standard candles. The book "Precision Cosmology" by Bernard Jones gives anecdotal evidence that as late as 1995 the answer to "What is the spatial geometry of the universe?" was still "up in the air". In the image below, different cosmologists give different answers to this question.

 

[RogerWaters]

Now I’m really confused. I thought you originally said that it’s not dark energy (by extension, density, dark energy making up the critical density) that causes flatness but cosmic inflation. Now I interpret you to be saying that mass/energy density causes flatness, in line with the second possibility wrote in the summary!

 

[PeroK]

Now I’m really confused. I thought you originally said that it’s not dark energy (by extension, density, dark energy making up the critical density) that causes flatness but cosmic inflation. Now I interpret you to be saying that mass/energy density causes flatness, in line with the second possibility wrote in the summary!

If you want to understand this beyond the popular science level, I would recommend An Introduction to Modern Cosmology by Andrew Liddle. Note that a knowledge of GR is not a prerequisite for this book.

1) Let's consider a universe without dark energy and without inflation.

We would expect the universe today to be curved, one way or the other. In order to be close to flat today, it would need to have been extraordinarily close to flat in the early universe. This seems too coincidental; too finely tuned. That's problem number 1: the fine tuning of total energy density in the early universe.

Let's accept, however, that the observations are correct: the universe is flat and expanding. Now, we run the Friedmann equation with the observed density of matter and (hypothesised) density of dark matter to check the expansion rate is consistent with the observed energy density. We find that it isn't! That's problem number 2. The current expansion rate requires more energy density that we have from matter (and a bit of radiation).

Finally, according to the Friedmann equation, the expansion of the universe should be slowing down, but it is in fact accelerating. That's problem number 3: accelerated expansion.

2) We can solve the second problem first. We add dark (vacuum) energy as required so that we have the critical energy density. Note that we have no specific theoretical justification for the magnitude of energy density of dark matter other than by cosmological observations. Ideally, of course, we would have some theory of what causes it and an independent calculation of its magnititude.

Dark energy solves problem number 2: the missing energy density.

3) Dark energy also solves problem number 3. As the universe expands, the density of matter and radiation decrease, but the density of dark energy remains constant (by its very nature). That's clear whether or not you're disturbed by the concept of negative pressure!

4) This leaves problem number 1: how did the energy density of the universe come to be fine-tuned? This is the problem that is solved by inflation (see the Guth video).

 

[Ibix]

Now I’m really confused. I thought you originally said that it’s not dark energy (by extension, density, dark energy making up the critical density) that causes flatness but cosmic inflation. Now I interpret you to be saying that mass/energy density causes flatness, in line with the second possibility wrote in the summary!

You need to be careful about language, because there are two related questions hiding here.

First, we can ask "is the universe spatially flat?", to which the answer appears to be yes. That, in turn, implies that there must be a certain density of matter and energy, because it's that density that causes the spatial curvature or lack thereof. Dark energy is a component of that, so you could say that dark energy is the "missing piece" that makes the density correct to cause flatness.

The second question is "why was the density exactly critical in the first place?" If it's even slightly off the critical value for flatness then the spatial curvature grows - and it's tiny now, so must have been ridiculously tiny in the past. Inflation is our answer to that one.

So the universe is flat because matter plus dark matter plus dark energy has a particular density. It has that density because of inflation.

 

[RogerWaters]

Thanks Ibix (andPeroK).

So inflation causes critical density, which in turn causes flatness.

How does inflation produce the right density of matter and energy so robustly? Is it possibly to relay the idea here in vague terms?

 

[PeterDonis]

The book "Precision Cosmology" by Bernard Jones gives anecdotal evidence that as late as 1995 the answer to "What is the spatial geometry of the universe?" was still "up in the air".

My statement may have been too strong regarding exact flatness. However, I still don't think the observations on spatial curvature were ever consistent with a density only 30% of critical. The 1995 opinion survey you refer to actually bears out that point, since the results obtained only make sense if the universe is close enough to flat that all three possibilities (open, flat, closed) are in play. That requires the density to be close enough to critical for all three possibilities to be in play. I don't think 30% of critical is anywhere near close enough for that (much less 5% of critical, which, as I mentioned before, is what all the matter we can actually see adds up to).

 

[PeterDonis]

I thought you originally said that it’s not dark energy (by extension, density, dark energy making up the critical density) that causes flatness but cosmic inflation. Now I interpret you to be saying that mass/energy density causes flatness

Read my post #10 again, carefully, particularly what it says about what "inflation" actually means.

 

[RogerWaters]

"Inflation" is just a shorthand name for a model that is dominated by a particular kind of stress-energy, which is a scalar field with a huge vacuum energy density. It's that property that causes inflationary expansion.

This? I'm not sure what you are suggesting I take from it by just saying 'read it carefully'.

Ibix related the causal sequence in a way which made sense to me, so if what he said is correct then it's all good.

I'm still interested, though, in how cosmic inflation was such a robust process for creating the critical energy/matter density. Books I've read by Krauss and Greene relate the idea as follows: if even a 3d-sphere were to rapidly expand, what was formally curved would be flat (or at least appear flat to anyone at any particular locality) - that's why I was under the erroneous impression that the critical density is a by-product of the flatness; the main causal agent, as it were, being the inflation.

Anyway I have Guth's book 'the inflationary universe' to read next.

 

[PeterDonis]

This?

Yes.

I'm not sure what you are suggesting I take from it by just saying 'read it carefully'.

The fact that it addresses the issue you raised. You were confused because I said "inflation" causes flatness, but then I said "mass/energy density" causes flatness. Those aren't different things; they're just two different ways of saying the same thing. "Inflation" is just a shorthand way of referring to a particular kind of mass/energy density (or stress-energy, as I called it).

Ibix related the causal sequence in a way which made sense to me, so if what he said is correct then it's all good.

What @Ibix said is fine as far as it goes, but you are correct that it still leaves an open question:

I'm still interested, though, in how cosmic inflation was such a robust process for creating the critical energy/matter density.

Inflation doesn't "create" the stress-energy density; the stress-energy density is already there during inflation, stored in the inflaton field. The process of inflation, exponential expansion at a huge rate, makes the stress-energy density stored in the inflaton field equal to the critical density--although it does that by changing the critical density, not the inflaton field density (since the latter is constant during the inflation epoch). So at the end of inflation, the critical density is equal to the stress-energy density in the inflaton field, and then that stress-energy density gets transferred from the inflaton field to the Standard Model fields (quarks, leptons, gauge bosons) in a process called "reheating" (which is a misnomer since the Standard Model fields had never been "heated" before that). Since the density was equal to critical before reheating, and reheating happens basically instantly, the density is equal to critical right after reheating.

 

[kimbyd]

The universe was observed to be spatially flat decades before dark energy was discovered. Even leaving aside all the issues I mentioned, if the universe is in fact spatially flat, or so close to it that we can't tell the difference, then the total energy density must be the critical density, or so close to it that we can't tell the difference. And "30% of critical density" certainly does not meet that requirement. So before dark energy was discovered, there was a huge disconnect between our observations of spatial flatness and our observations of energy density. The discovery of dark energy solved that disconnect, but left other questions still unresolved.

I wouldn't say that exactly. The observations of a near-flat universe today weren't made until right around the same time as dark energy.

Rather, it was known that even if the universe wasn't flat, it had to be fairly close to it in order to have lasted even a few billion years. Too negative and no structure would have formed. Too positive and the universe would have quickly recollapsed. For neither to occur, the curvature in the early universe had to be very, very nearly flat. With this reasoning, there could still be quite enough curvature to be measurable. But the flatness still needed explaining even then.

Inflation is one solution to this because it drives the universe to be exponentially-close to perfect flatness. In most inflation models, no measurable deviation from flatness is predicted.

 

[PeterDonis]

it was known that even if the universe wasn't flat, it had to be fairly close to it in order to have lasted even a few billion years. Too negative and no structure would have formed. Too positive and the universe would have quickly recollapsed.

Yes, good point. And a density only 30% of critical (much less only 5% of critical) would not be anywhere close enough to critical for this.

 

[RogerWaters]

Yes.The fact that it addresses the issue you raised. You were confused because I said "inflation" causes flatness, but then I said "mass/energy density" causes flatness. Those aren't different things; they're just two different ways of saying the same thing. "Inflation" is just a shorthand way of referring to a particular kind of mass/energy density (or stress-energy, as I called it).What @Ibix said is fine as far as it goes, but you are correct that it still leaves an open question:Inflation doesn't "create" the stress-energy density; the stress-energy density is already there during inflation, stored in the inflaton field. The process of inflation, exponential expansion at a huge rate, makes the stress-energy density stored in the inflaton field equal to the critical density--although it does that by changing the critical density, not the inflaton field density (since the latter is constant during the inflation epoch). So at the end of inflation, the critical density is equal to the stress-energy density in the inflaton field, and then that stress-energy density gets transferred from the inflaton field to the Standard Model fields (quarks, leptons, gauge bosons) in a process called "reheating" (which is a misnomer since the Standard Model fields had never been "heated" before that). Since the density was equal to critical before reheating, and reheating happens basically instantly, the density is equal to critical right after reheating.

Thanks - that's very helpful.

These are no doubt obtuse questions I'm asking, but I hope questions coming from non-physicists are not deemed too excruciating on these boards.

 

[RogerWaters]

The observations of a near-flat universe today weren't made until right around the same time as dark energy.

Was this the BOOMERanG experiment?

 

[PeterDonis]

These are no doubt obtuse questions I'm asking

Not at all. Nobody is born understanding any physics topic. We all have to learn at some point.

I hope questions coming from non-physicists are not deemed too excruciating on these boards.

Certainly not. In fact one of PF's main purposes is to help non-physicists understand physics.

 

[PeroK]

These are no doubt obtuse questions I'm asking, but I hope questions coming from non-physicists are not deemed too excruciating on these boards.

On the contrary, it's a pleasure to answer your questions. Take a look a this thread if you want to see how excrutiating it becomes at times:

https://www.physicsforums.com/threads/can-the-dark-energy-concept-be-wrong.1011169/

 

[George Jones]

Yes, good point. And a density only 30% of critical (much less only 5% of critical) would not be anywhere close enough to critical for this.

But this is where things stood in 80s, before dark energy in the late 90s. From page 18
https://arxiv.org/abs/2201.04741

"By 1983, inflation had become the driving force in cosmology because of the power of its three big predictions: (i) flat Universe ($\Omega_0=1$); (ii) almost scale-invariant spectrum of nearly-Gaussian density (curvature) perturbations;21 and (iii) almost scale-invariant spectrum of gravitational waves (134).

The first prediction made inflation both bold and falsifiable: the observational evidence at the time was $\Omega_0 \sim 0.1$, and so there must be something in the Universe in addition to baryons"

The article at the link above is a detailed history of modern cosmology by one of giants, Mike Turner, and is technical but largely non-mathematical. I "discovered" it a couple of hours ago.

 

[PeterDonis]

the observational evidence at the time was $\Omega_0 \sim 0.1$, and so there must be something in the Universe in addition to baryons"

Yes, but even without inflation, we already knew that the total density had to be more than that. If that really were all that was in the universe, then, as @kimbyd pointed out, no structure would have formed. So while inflation certainly did predict that "there must be something in the Universe in addition to baryons", I don't think inflation was necessary to obtain that particular prediction--just the observation that there is structure in the universe was sufficient for that.

 

[RogerWaters]

Inflation doesn't "create" the stress-energy density; the stress-energy density is already there during inflation, stored in the inflaton field. The process of inflation, exponential expansion at a huge rate, makes the stress-energy density stored in the inflaton field equal to the critical density--although it does that by changing the critical density, not the inflaton field density (since the latter is constant during the inflation epoch). So at the end of inflation, the critical density is equal to the stress-energy density in the inflaton field, and then that stress-energy density gets transferred from the inflaton field to the Standard Model fields (quarks, leptons, gauge bosons) in a process called "reheating" (which is a misnomer since the Standard Model fields had never been "heated" before that). Since the density was equal to critical before reheating, and reheating happens basically instantly, the density is equal to critical right after reheating.

I've been reading this over a few times. How did inflation change the critical density, such that it become equal to the mass-density of the universe (which was then the stress-energy density of the inflaton field)?

As a side note, you say (and I have commonly read) that the stress-energy stored in the inflaton field got transferred into matter at the end of inflation. Given matter (including dark) represents around 30 percent of the mass-energy of the universe, does this mean only 30 percent of the inflaton stress-energy turned into matter? And similarly did 70 percent of the inflaton stress-energy come to be dark energy (I understand that as space expands, dark energy density remains constant and so I assume its contribution to the mass-density of the universe was there from the beginning)?

 

[PeterDonis]

How did inflation change the critical density, such that it become equal to the mass-density of the universe (which was then the stress-energy density of the inflaton field)?

Because that's what that kind of stress-energy (i.e., any stress-energy that has the same equation of state as a cosmological constant, with $p = - \rho$) does. In fact, in de Sitter spacetime, which is a spacetime with a positive cosmological constant and no other stress-energy, the actual density (i.e., the density associated with the cosmological constant) and the critical density are always exactly equal. Any stress-energy with the same equation of state as a cosmological constant rapidly pushes the critical density towards the same equality with the actual density that exists all the time in de Sitter spacetime.

 

[RogerWaters]

Because that's what that kind of stress-energy (i.e., any stress-energy that has the same equation of state as a cosmological constant, with $p = - \rho$) does. In fact, in de Sitter spacetime, which is a spacetime with a positive cosmological constant and no other stress-energy, the actual density (i.e., the density associated with the cosmological constant) and the critical density are always exactly equal. Any stress-energy with the same equation of state as a cosmological constant rapidly pushes the critical density towards the same equality with the actual density that exists all the time in de Sitter spacetime.

I'm sure it just comes out of the math that way, but are there any analogies or thought experiments that might help one understand why inflation does this? I am supposing the critical density for flatness to be a mass-energy (or stress-energy) density of the universe (or inflaton field) which balances the rate of expansion such that curvature does not occur (but this might be conflating the critical density for expansion vs collapse with the critical density for curvature). It boggles me that the nature of that stress-energy could change the amount of mass/energy needed for flatness, as opposed to being an input into whether the universe is, in fact, flat or not given a certain expansion rate (if that makes sense) .

Also I edited by last post to include this further question, but probably after you saw it:

As a side note, you say (and I have commonly read) that the stress-energy stored in the inflaton field got transferred into matter at the end of inflation. Given matter (including dark) represents around 30 percent of the mass-energy of the universe, does this mean only 30 percent of the inflaton stress-energy turned into matter? And similarly did 70 percent of the inflaton stress-energy come to be dark energy (I understand that as space expands, dark energy density remains constant and so I assume its contribution to the mass-density of the universe was there from the beginning)?

 

[PeterDonis]

are there any analogies or thought experiments that might help one understand why inflation does this?

Do you understand why de Sitter spacetime (i.e., positive cosmological constant and no other stress-energy) has the actual density equal to the critical density always? I would suggest thinking about that first. Once you understand that, the idea that that de Sitter state is a "fixed point" towards which any inflation model will drive the universe should be pretty easy to grasp.

 

[RogerWaters]

Do you understand why de Sitter spacetime (i.e., positive cosmological constant and no other stress-energy) has the actual density equal to the critical density always? I would suggest thinking about that first. Once you understand that, the idea that that de Sitter state is a "fixed point" towards which any inflation model will drive the universe should be pretty easy to grasp.

Nope not at all.

 

[PeterDonis]

I am supposing the critical density for flatness to be a mass-energy (or stress-energy) density of the universe (or inflaton field) which balances the rate of expansion such that curvature does not occur (but this might be conflating the critical density for expansion vs collapse with the critical density for curvature).

If the only stress-energy in the universe is matter ($p = 0$) or radiation ($p = \rho / 3$), then the two concepts of "critical density" (expansion vs. collapse and zero spatial curvature) coincide. But in the presence of stress-energy with the equation of state of a cosmological constant ($p = - \rho$), they don't. What cosmologists call the "critical density" for our best current model of the universe, which includes a positive cosmological constant, is the "flatness" one, although many cosmologists are not clear about that and will refer to the "expansion vs. collapse" definition without clarifying that that definition doesn't really apply to our actual best current model of the universe. (Believe it or not, Wikipedia actually gets this right in its "critical density" article.)

 

[PeterDonis]

Nope not at all.

Mathematically it's easily seen. The critical density is $\rho_c = 3 H^2 / 8 \pi$, and the first Friedmann equation for the de Sitter case says $H^2 = \Lambda / 3$. To convert $\Lambda$ to its equivalent density we have $\rho_\Lambda = \Lambda / 8 \pi$, so we have $H^2 = 8 \pi \rho_\Lambda / 3$. Plugging that into the $\rho_c$ equation gives $\rho_c = \rho_\Lambda$.

Physically, the best intuition I know of is the one usually given for inflation models, that exponential expansion dilutes spatial curvature, and the greater the exponential factor the greater the dilution. Since typical inflation models have some 60 or so e-foldings, i.e., an exponential factor of $e^{60}$, which is huge, we would expect them to hugely dilute any pre-existing spatial curvature. So the fixed point of exponential expansion would be expected to be zero spatial curvature, i.e., actual density equal to critical density. And that is what we see in de Sitter spacetime, which is the fixed point of exponential expansion.

 

[RogerWaters]

Physically, the best intuition I know of is the one usually given for inflation models, that exponential expansion dilutes spatial curvature, and the greater the exponential factor the greater the dilution. Since typical inflation models have some 60 or so e-foldings, i.e., an exponential factor of $e^{60}$, which is huge, we would expect them to hugely dilute any pre-existing spatial curvature. So the fixed point of exponential expansion would be expected to be zero spatial curvature, i.e., actual density equal to critical density. And that is what we see in de Sitter spacetime, which is the fixed point of exponential expansion.

Right, I read this frequently and understand the intuition as far as resulting in flatness but not mass-energy density being at the critical value. I appreciate now that these are one and the same thing, as opposed to critical mass-energy density being a side effect. However, exponential expansion of, say, a massive balloon may all but flatten local sections of it (to an observer living on the surface) but it won’t change the critical density needed for flatness (I don’t think?)- I guess this is where analogy breaks down and you need to do the physics.